Free Printable M7

1.

What is the value of $e^{\left(i\pi\right)}$ ?

a)

2

b)

0

c)

1

d)

-1

2.

Simplify $e^{\left(iθ\right)}$ in terms of sine and cosine functions.

a)

$\sin(θ)+i\cos(θ)$

b)

$\cos(θ)+i\sin(θ)$

c)

$\csc(θ)+i\cot(θ)$

d)

$\tan(θ)+i\sec(θ)$

3.

Express the hyperbolic sine function in terms of exponential functions.

a)

$\frac{\left(e^z-e^{\left(-z\right)}\right)}{2}$

b)

$\left(e^z-e^{\left(-z\right)}\right)$

c)

$\frac{\left(e^z+e^{\left(-z\right)}\right)}{2}$

d)

$\frac{\left(e^z.e^{\left(-z\right)}\right)}{2}$

4.

What is the principal value of the complex logarithm of z?

a)

Modulus of z plus i times the argument of z

b)

Natural logarithm of z

c)

Natural logarithm of the modulus of z plus i times the argument of z

d)

Logarithm of the conjugate of z

5.

What is the value of $(1+i)^2$ ?

a)

$2i$

b)

$i$

c)

1

d)

0

6.

Calculate the value of $(5-2i)^3$ .

a)

$125-100i$

b)

$65+142i$

c)

$117-44i$

d)

$120-80i$

7.

When will we say that a function f(z) is a periodic function?

a)

f(z+c)=f(c) for all z

b)

f(z+c)=f(z) for all z

c)

f(z+c)=f(-z) for all z

d)

f(z+c)=f(c) for all c

8.

The distance between z=4-3i and w=-2-4i on the complex plane is...

a)

√35

b)

√35 i

c)

√37

d)

5i

9.

Let $w=-\frac{\sqrt{3}}{2}+\frac{1}{2}i$ . Calculate the modulus and argument.

a)

3, $\frac{2}{3}\pi$

b)

5, $\frac{3}{2}\pi$

c)

1, $\frac{5}{6}\pi$

d)

Not Possible

10.

If z=√2+√2 i then arg(z²)=

a)

0

b)

$\frac{\pi}{3}$

c)

$\frac{\pi}{2}$

d)

Not Possible

11.

$Write\ in\ Polar\ Form\ :$ $3\ +6i$

a)

$3\sqrt{5\ }\left(Cos\ \frac{\pi}{3}+\ i\ \sin\ \frac{\pi}{3}\right)$

b)

$3\sqrt{5}\left(Cos\ 1.124\ \ +\ i\ \sin\ 1.124\right)$

c)

$3\sqrt{5}\left(Cos\ 1.107\ +\ i\ Sin\ 1.107\right)$

d)

$3\sqrt{5}\left(Cos\ 1.141\ +\ i\ Sin\ 1.141\right)$

12.

$WRITE\ IN\ TRIGONOMETRIC\ FORM:\ \ \$ $-1\ +i$

a)

$\left(\sqrt{2}\left(\cos\left(\frac{1}{4}\pi\right)\right)\ +\ i\ \sin\ \left(\frac{1}{4}\pi\right)\right)$

b)

$\sqrt{2}\left(\cos\ \frac{3}{4}\pi\ +i\ \sin\ \frac{3}{4}\pi\right)$

c)

$\sqrt{3}\left(\sin\ \frac{1}{4}\pi+\ \cos\ \frac{1}{4}\pi\right)$

d)

$\sqrt{2}\left(\cos\ \frac{5}{4}\pi\ +\ \sin\ \frac{5}{4}\pi\right)$

13.

What is the Modulus in a complex number $1\ +\ i\sqrt{3}$

a)

2

b)

4

c)

$\frac{\pi}{4}$

d)

$\frac{\pi}{3}$

14.

Write the complex number in polar form

a)

3 ( cos 3π/2 + i sin 3π/2)

b)

3 ( cos π/2 + i sin π/2)

c)

9 ( cos 3π/2 + i sin 3π/2)

d)

3 ( cos 2π + i 2π)

15.

In $\left(9+3i\right)$ , the imaginary part is (a) .

16.

How can you identify a complex unit in a graph?

a)

By seeing if it is in cartesian coordinate and Y-axis number has an "i"

b)

By seeing if it is in polar coordinate and Y-axis number has an "i"

c)

By asking to your teacher the right answer

17.

Express the point in trig form in standard form.
4 ( cos π /3 + i sin π /3)

a)

2 + 2√3 i

b)

2 + √3 i

c)

2 - 2√3 i

d)

-2 + 2√3 i

18.

If $z = 5 \text{cis} \left(\frac{\pi}{3}\right)$ , find $z^2$ using DeMoivre's Theorem.

a)

$25 \text{cis} \left(\frac{2\pi}{3}\right)$

b)

$25 \text{cis} \left(\frac{\pi}{3}\right)$

c)

$10 \text{cis} \left(\frac{2\pi}{3}\right)$

d)

$25 \text{cis} \left(\pi\right)$

19.

What is the result of multiplying $3 \text{cis} \left(\frac{\pi}{4}\right)$ by $2 \text{cis} \left(\frac{\pi}{6}\right)$ ?

a)

$6 \text{cis} \left(\frac{5\pi}{12}\right)$

b)

$6 \text{cis} \left(\frac{\pi}{3}\right)$

c)

$5 \text{cis} \left(\frac{\pi}{2}\right)$

d)

$6 \text{cis} \left(\frac{5\pi}{6}\right)$

20.

Using DeMoivre's Theorem, calculate $(1+i)^6$ .

a)

$-8-8i$

b)

$8i$

c)

$-8i$

d)

$8-8i$

21.

Find the sixth roots of $64 \text{cis} (\pi)$ .

a)

$2 \text{cis} \left(\frac{\pi}{6}\right)$ , $2 \text{cis} \left(\frac{\pi}{2}\right)$ , $2 \text{cis} \left(\frac{5\pi}{6}\right)$ , $2 \text{cis} \left(\frac{7\pi}{6}\right)$ , $2 \text{cis} \left(\frac{3\pi}{2}\right)$ , $2 \text{cis} \left(\frac{11\pi}{6}\right)$

b)

$2 \text{cis} \left(\frac{\pi}{6}\right)$ , $2 \text{cis} \left(\frac{\pi}{3}\right)$ , $2 \text{cis} \left(\frac{\pi}{2}\right)$ , $2 \text{cis} \left(\frac{2\pi}{3}\right)$ , $2 \text{cis} \left(\frac{5\pi}{6}\right)$ , $2 \text{cis} \left(\pi\right)$

c)

$2 \text{cis} \left(\frac{\pi}{6}\right)$ , $2 \text{cis} \left(\frac{\pi}{3}\right)$ , $2 \text{cis} \left(\frac{\pi}{2}\right)$ , $2 \text{cis} \left(\frac{2\pi}{3}\right)$ , $2 \text{cis} \left(\frac{5\pi}{6}\right)$ , $2 \text{cis} \left(\frac{7\pi}{6}\right)$

d)

$2 \text{cis} \left(\frac{\pi}{6}\right)$ , $2 \text{cis} \left(\frac{\pi}{3}\right)$ , $2 \text{cis} \left(\frac{\pi}{2}\right)$ , $2 \text{cis} \left(\frac{2\pi}{3}\right)$ , $2 \text{cis} \left(\frac{5\pi}{6}\right)$ , $2 \text{cis} \left(\frac{11\pi}{6}\right)$

22.

If $z = 4 \text{cis} \left(\frac{\pi}{6}\right)$ , what is $z^4$ using DeMoivre's Theorem?

a)

$256 \text{cis} \left(\frac{2\pi}{3}\right)$

b)

$256 \text{cis} \left(\frac{\pi}{6}\right)$

c)

$256 \text{cis} \left(\frac{4\pi}{3}\right)$

d)

$256 \text{cis} \left(\frac{2\pi}{3}\right)$

23.

What are the cube roots of $8 \text{cis} (0)$ ?

a)

$2 \text{cis} \left(0\right)$ , $2 \text{cis} \left(\frac{2\pi}{3}\right)$ , $2 \text{cis} \left(\frac{4\pi}{3}\right)$

b)

$2 \text{cis} \left(0\right)$ , $2 \text{cis} \left(\frac{\pi}{3}\right)$ , $2 \text{cis} \left(\frac{2\pi}{3}\right)$

c)

$2 \text{cis} \left(0\right)$ , $2 \text{cis} \left(\pi\right)$ , $2 \text{cis} \left(2\pi\right)$

d)

$2 \text{cis} \left(0\right)$ , $2 \text{cis} \left(\frac{\pi}{2}\right)$ , $2 \text{cis} \left(\pi\right)$

24.

$x^2-2x+3=0$

What are the roots of the given quadratic equation?

a)

$x=3\ or\ x=-1$

b)

$x=1+\sqrt{2}i\ or\ x=1-\sqrt{2}i$

c)

$x=2+\sqrt{2}i\ or\ x=2-\sqrt{2}i$

d)

$x=1+2i\ or\ x=1-2i$

25.

Sabiendo que $f\left(z\right)=z^2+1$ , calcule el valor de

$f\left(i\right)$

a)

0

b)

1

c)

i

d)

-i

26.

Sabiendo que $f\left(x+yi\right)=2x-y+3xi$ , determine el valor de

$K=f\left(1+i\right)$

a)

1+3i

b)

2+3i

c)

3+2i

d)

2

M7 - EP1 2025

Total questions: 26

Worksheet time: 48mins

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